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We present a detailed stellar population analysis of 11 bright ($H<26.6$) galaxies at $z=9-11$ (three spectroscopically confirmed) to constrain the chemical enrichment and growth of stellar mass of early galaxies. We use the flexible Bayesian spectral energy distribution (SED) fitting code Prospector with a range of star-formation histories (SFHs), a flexible dust attenuation law and a self-consistent modeling of emission lines. This approach allows us to assess how different priors affect our results, and how well we can break degeneracies between dust attenuation, stellar ages, metallicity and emission lines using data which probe only the rest-frame ultraviolet to optical wavelengths. We measure a median observed ultraviolet spectral slope $\beta=-1.87_{-0.43}^{+0.35}$ for relatively massive star-forming galaxies ($9<\log(M_{\star}/M_{\odot})<10$), consistent with no change from $z=4$ to $z=9-10$ at these stellar masses, implying rapid enrichment. Our SED-fitting results are consistent with a star-forming main sequence with sub-linear slope ($0.7\pm0.2$) and specific star-formation rates of $3-10~\mathrm{Gyr}^{-1}$. However, the stellar ages and SFHs are less well constrained. Using different SFH priors, we cannot distinguish between median mass-weighted ages of $\sim50-150$ Myr, which corresponds to 50\% formation redshifts of $z_{50}\sim10-12$ at $z\sim9$ and is of the order of the dynamical timescales of these systems. Importantly, the models with different SFH priors are able to fit the data equally well. We conclude that the current observational data cannot tightly constrain the mass-buildup timescales of these $z=9-11$ galaxies, with our results consistent with SFHs implying both a shallow and steep increase of the cosmic SFR density with time at $z>10$.